1. Reference problem#

1.1. Geometry#

We consider a square plate with side \(1m\) and thickness \(\mathrm{0,2}m\).

1.2. Material properties#

Nil.

1.3. Boundary conditions and loads#

A mechanical resolution operator is not called; generalized analytical effort fields are given at the input of CALC_FERRAILLAGE, corresponding to one of the following configurations:

1.3.1. Case of loading at ELU#

  1. compression force of \(1000000\text{N}\) exerted along the \(\text{Y}\) axis, and a shear force of \(100000\text{N}\) along \(\text{Y}\)

  2. traction force of \(1000000\text{N}\) exerted along the \(\text{X}\) axis, and a shear force of \(-600000\text{N}\) along \(\text{X}\)

  3. tractive force of \(1000000\text{N}\) exerted along the \(\text{Y}\) axis, and a shear force of \(-20000\text{N}\) along \(\text{X}\) and \(80000\text{N}\) along \(\text{Y}\)

  4. bending moment of \(100000\text{Nm}\) around the \(\text{Y}\) axis.

  5. Bending moment of \(100000\text{Nm}\) around the \(\text{X}\) axis.

  6. bending moment of \(100000\text{Nm}\) around the \(\text{X}\) axis and \(100000\text{N}\) compression force exerted along the \(\text{X}\) axis

  7. bending moment of \(100000\text{Nm}\) around the \(\text{X}\) axis and \(100000\text{N}\) traction force exerted along the \(\text{X}\) axis

  8. bending moment of \(100000\text{Nm}\) around the \(\text{X}\) axis and \(2000000\text{N}\) traction force exerted along the \(\text{X}\) axis

  9. bending moment of \(100000\text{Nm}\) around the \(\text{X}\) axis and \(-75000\text{Nm}\) bending moment around the \(\text{Y}\) axis

  10. bending moment of \(-150000\text{Nm}\) around the \(\text{Y}\) axis.

  11. bending moment of \(-260000\text{Nm}\) around the \(\text{Y}\) axis.

  12. bending moment of \(-380000\text{Nm}\) around the \(\text{Y}\) axis.

  13. compression force of \(1500000\text{N}\) exerted along the \(\text{Y}\) axis, and a shear force of \(800000\text{N}\) along \(\text{Y}\)

  14. bending moment of \(380000\text{Nm}\) around the axis \(\text{X}\), compression force of \(4500000\text{N}\) exerted along the axis \(X\), and a shear force of \(100000\text{N}\) following \(\text{Y}\)

1.3.2. Load cases with the ELS Feature#

  1. compression force of \(1000000\text{N}\) exerted along the \(\text{Y}\) axis, and a shear force of \(100000\text{N}\) along \(\text{Y}\)

  2. traction force of \(1000000\text{N}\) exerted along the \(\text{X}\) axis, and a shear force of \(-600000\text{N}\) along \(\text{X}\)

  3. tractive force of \(1000000\text{N}\) exerted along the \(\text{Y}\) axis, and a shear force of \(-20000\text{N}\) along \(\text{X}\) and \(80000\text{N}\) along \(\text{Y}\)

  4. bending moment of \(100000\text{Nm}\) around the \(\text{Y}\) axis.

  5. Bending moment of \(100000\text{Nm}\) around the \(\text{X}\) axis.

  6. bending moment of \(300000\text{Nm}\) around the \(\text{X}\) axis and \(20000\text{N}\) compression force exerted along the \(\text{X}\) axis

  7. bending moment of \(100000\text{Nm}\) around the \(\text{X}\) axis and \(100000\text{N}\) traction force exerted along the \(\text{X}\) axis

  8. bending moment of \(100000\text{Nm}\) around the \(\text{X}\) axis and \(2000000\text{N}\) traction force exerted along the \(\text{X}\) axis

  9. bending moment of \(100000\text{Nm}\) around the \(\text{X}\) axis and \(-75000\text{Nm}\) bending moment around the \(\text{Y}\) axis

  10. bending moment of \(-300000\text{Nm}\) around the \(\text{Y}\) axis.

1.3.3. Case of loading at the ELS Quasi-Permanent#

  1. compression force of \(1000000\text{N}\) exerted along the \(\text{Y}\) axis, and a shear force of \(100000\text{N}\) along \(\text{Y}\)

  2. traction force of \(1000000\text{N}\) exerted along the \(\text{X}\) axis, and a shear force of \(-600000\text{N}\) along \(\text{X}\)

  3. tractive force of \(1000000\text{N}\) exerted along the \(\text{Y}\) axis, and a shear force of \(-20000\text{N}\) along \(\text{X}\) and \(80000\text{N}\) along \(\text{Y}\)

  4. bending moment of \(100000\text{Nm}\) around the \(\text{Y}\) axis.

  5. Bending moment of \(100000\text{Nm}\) around the \(\text{X}\) axis.

  6. bending moment of \(300000\text{Nm}\) around the \(\text{X}\) axis and \(15000\text{N}\) compression force exerted along the \(\text{X}\) axis

  7. bending moment of \(100000\text{Nm}\) around the \(\text{X}\) axis and \(100000\text{N}\) traction force exerted along the \(\text{X}\) axis

  8. bending moment of \(100000\text{Nm}\) around the \(\text{X}\) axis and \(2000000\text{N}\) traction force exerted along the \(\text{X}\) axis

  9. bending moment of \(100000\text{Nm}\) around the \(\text{X}\) axis and \(-75000\text{Nm}\) bending moment around the \(\text{Y}\) axis

  10. bending moment of \(-125000\text{Nm}\) around the \(\text{Y}\) axis.

1.4. Other calculation parameters#

1.4.1. Settings at ELU#

In ELU, we will consider 9 configurations on which we will start the calculation of the 14 load cases presented in §1.3.1:

  • Setup 1- Basic calculation:

  • calculation at EC2

  • upper coating of csup = 4 cm

  • lower cinf coating = 4 cm

  • elastic limit of steel fyk= 500 MPa

  • Young’s modulus of Eys steel = 210,000 MPa

  • type diagram (sant-901) for steel = “B2”

  • characteristic strength of concrete fck= 35 MPa

  • density of the steel = 7800 Kg/m3

  • αcc= 1.0/γc= 1.5/γs= 1.15

  • Configuration 2- Taking into account compression reinforcement:

  • calculation at EC2

  • FERR_COMP = “OUI “

  • upper coating of csup = 4 cm

  • lower cinf coating = 4 cm

  • elastic limit of steel fyk= 500 MPa

  • Young’s modulus of Eys steel = 210,000 MPa

  • type diagram (sant-901) for steel = “B2”

  • characteristic strength of concrete fck= 35 MPa

  • density of the steel = 7800 Kg/m3

  • αcc= 1.0/γc= 1.5/γs= 1.15

  • Configuration 3- Taking into account the minimum reinforcement see EC2:

  • calculation at EC2

  • FERR_COMP = “OUI “

  • FERR_MIN = “CODE “

  • upper coating of csup = 4 cm

  • lower cinf coating = 4 cm

  • elastic limit of steel fyk= 500 MPa

  • Young’s modulus of Eys steel = 210,000 MPa

  • type diagram (sant-901) for steel = “B2”

  • characteristic strength of concrete fck= 35 MPa

  • density of the steel = 7800 Kg/m3

  • αcc= 1.0/γc= 1.5/γs= 1.15

  • Configuration 4- Taking into account a minimum reinforcement provided by the user:

  • calculation at EC2

  • FERR_COMP = “OUI “

  • FERR_MIN = “OUI “

  • RHO_LONGI_MIN = 0.0013,

  • RHO_TRNSV_MIN = 0. ,

  • upper coating of csup = 4 cm

  • lower cinf coating = 4 cm

  • elastic limit of fyk steel = 500 MPa

  • Young’s modulus of Eys steel = 210,000 MPa

  • type diagram (sant-901) for steel = “B2”

  • characteristic strength of concrete fck = 35 MPa

  • density of the steel [s] = 7800 kg/m3

  • αcc = 1.0/γc = 1.5/γs = 1.15

  • Configuration 5- Change in the type of the diagram (ys−e) for steel:

  • calculation at EC2

  • FERR_COMP = “OUI “

  • FERR_MIN = “CODE “

  • upper coating of csup = 4 cm

  • lower cinf coating = 4 cm

  • elastic limit of steel fyk= 500 MPa

  • Young’s modulus of Eys steel = 210,000 MPa

  • type diagram (sant-901) for steel = “B1”

  • characteristic strength of concrete fck= 35 MPa

  • density of the steel = 7800 Kg/m3

  • αcc= 1.0/γc= 1.5/γs= 1.15

  • Configuration 6- Taking into account the impact of compression on the shear force resistance:

  • calculation at EC2

  • FERR_COMP = “OUI “

  • FERR_MIN = “CODE “

  • UTIL_COMPR = “OUI “

  • upper coating of csup = 4 cm

  • lower cinf coating = 4 cm

  • elastic limit of steel fyk= 500 MPa

  • Young’s modulus of Eys steel = 210,000 MPa

  • type diagram (sant-901) for steel = “B2”

  • characteristic strength of concrete fck= 35 MPa

  • density of the steel = 7800 Kg/m3

  • αcc= 1.0/γc= 1.5/γs= 1.15

  • Configuration 7- Taking into account the impact of shear force and torsion on the longitudinal reinforcement:

  • calculation at EC2

  • FERR_COMP = “OUI “

  • FERR_MIN = “CODE “

  • EPURE_CISA = “OUI “

  • upper coating of csup = 4 cm

  • lower cinf coating = 4 cm

  • elastic limit of steel fyk= 500 MPa

  • Young’s modulus of Eys steel = 210,000 MPa

  • type diagram (sant-901) for steel = “B2”

  • characteristic strength of concrete fck= 35 MPa

  • density of the steel = 7800 Kg/m3

  • αcc= 1.0/γc= 1.5/γs= 1.15

  • Configuration 8- Search for a symmetric sizing reinforcement:

  • calculation at EC2

  • FERR_COMP = “OUI “

  • FERR_SYME = “OUI “

  • SEUIL_SYME = 1 cm2

  • upper coating of csup = 4 cm

  • lower cinf coating = 4 cm

  • elastic limit of steel fyk= 500 MPa

  • Young’s modulus of Eys steel = 210,000 MPa

  • type diagram (sant-901) for steel = “B2”

  • characteristic strength of concrete fck= 35 MPa

  • density of the steel = 7800 Kg/m3

  • αcc= 1.0/γc= 1.5/γs= 1.15

  • Configuration 9- Variation of the search threshold for a symmetric sizing reinforcement:

  • calculation at EC2

  • FERR_COMP = “OUI “

  • FERR_SYME = “OUI “

  • SEUIL_SYME = 10 cm2

  • upper coating of csup = 4 cm

  • lower cinf coating = 4 cm

  • elastic limit of steel fyk= 500 MPa

  • Young’s modulus of Eys steel = 210,000 MPa

  • type diagram (sant-901) for steel = “B2”

  • characteristic strength of concrete fck= 35 MPa

  • density of the steel = 7800 Kg/m3

  • αcc= 1.0/γc= 1.5/γs= 1.15

  • Setup 10- Sizing at BAEL91:

  • calculation at BAEL91

  • upper coating of csup = 4 cm

  • lower cinf coating = 4 cm

  • elastic limit of steel Fe= 500 MPa

  • Young’s modulus of Eys steel = 210,000 MPa

  • diagram type (S-E) for steel = “B2”

  • characteristic strength of concrete FCj= 35 MPa

  • density of the steel = 7800 Kg/m3

  • αcc= 1.0/γc= 1.5/γs= 1.15

  • Configuration 11- Change of steel class (A):

  • calculation at EC2

  • FERR_COMP = “OUI “

  • CLASSE_ACIER = “A”

  • upper coating of csup = 4 cm

  • lower cinf coating = 4 cm

  • elastic limit of fyk steel = 500 MPa

  • Young’s modulus of Eys steel = 210,000 MPa

  • diagram type (S-E) for steel = “B2”

  • characteristic strength of concrete fck = 35 MPa

  • density of the steel [s] = 7800 kg/m3

  • αcc = 1.0/γc = 1.5/γs = 1.15

  • Configuration 12- Steel class change (C):

  • calculation at EC2

  • FERR_COMP = “OUI “

  • CLASSE_ACIER = “C”

  • upper coating of csup = 4 cm

  • lower cinf coating = 4 cm

  • elastic limit of fyk steel = 500 MPa

  • Young’s modulus of Eys steel = 210,000 MPa

  • diagram type (S-E) for steel = “B2”

  • characteristic strength of concrete fck = 35 MPa

  • density of the steel [s] = 7800 kg/m3

  • αcc = 1.0/γc = 1.5/γs = 1.15

  • Configuration 13- Calculation of reinforcement with the “SANDWICH” method:

  • calculation at EC2

  • FERR_COMP = “OUI “

  • METHODE_2D = “Sandwich”

  • upper coating of csup = 4 cm

  • lower cinf coating = 4 cm

  • elastic limit of fyk steel = 500 MPa

  • Young’s modulus of Eys steel = 210,000 MPa

  • diagram type (S-E) for steel = “B2”

  • characteristic strength of concrete fck = 35 MPa

  • density of the steel [s] = 7800 kg/m3

  • αcc = 1.0/γc = 1.5/γs = 1.15

  • Configuration 14- Calculation of reinforcement with the method “SANDWICH “+ Variation of precision criteria:

  • calculation at EC2

  • FERR_COMP = “OUI “

  • METHODE_2D = “Sandwich”

  • THETA_ITER = 1.0

  • EP_ITER = 0.02

  • ALPHA_ITER = 0.15

  • upper coating of csup = 4 cm

  • lower cinf coating = 4 cm

  • elastic limit of fyk steel = 500 MPa

  • Young’s modulus of Eys steel = 210,000 MPa

  • diagram type (S-E) for steel = “B2”

  • characteristic strength of concrete fck = 35 MPa

  • density of the steel [s] = 7800 kg/m3

  • αcc = 1.0/γc = 1.5/γs = 1.15

  • Configuration 15- Calculation of reinforcement with the method “SANDWICH “+ Not taken into account §6.109 of EN-1992-2:

  • calculation at EC2

  • FERR_COMP = “OUI “

  • METHODE_2D = “Sandwich”

  • COND_109 = “NON “

  • upper coating of csup = 4 cm

  • lower cinf coating = 4 cm

  • elastic limit of fyk steel = 500 MPa

  • Young’s modulus of Eys steel = 210,000 MPa

  • diagram type (S-E) for steel = “B2”

  • characteristic strength of concrete fck = 35 MPa

  • density of the steel [s] = 7800 kg/m3

  • αcc = 1.0/γc = 1.5/γs = 1.15

1.4.2. Parameters with the ELS Feature#

In ELS Characteristic, we will consider 4 configurations on which we will start the calculation of the 10 load cases presented in §1.3.2:

  • Setup 1- Basic calculation:

  • calculation at EC2

  • upper coating of csup = 4 cm

  • lower cinf coating = 4 cm

  • characteristic strength of concrete fck= 35 MPa

  • authorized limit stress in steel: μs, lim = 400 MPa

  • maximum compressive stress authorized at the level of the underside of concrete:

►c, inf, lim = 21 MPa

  • maximum compressive stress authorized at the level of the underside of concrete:

►c, sup, lim=21 MPa

  • steel-concrete equivalence coefficient: αE = 21

  • density of the steel [s] = 7800 Kg/m3

  • Configuration 2- Taking into account compression reinforcement:

  • calculation at EC2

  • FERR_COMP = “OUI “

  • upper coating of csup = 4 cm

  • lower cinf coating = 4 cm

  • characteristic strength of concrete fck= 35 MPa

  • authorized limit stress in steel: μs, lim = 400 MPa

  • maximum compressive stress authorized at the level of the underside of concrete:

►c, inf, lim = 21 MPa

  • maximum compressive stress authorized at the level of the underside of concrete:

►c, sup, lim=21 MPa

  • steel-concrete equivalence coefficient: αE = 21

  • density of the steel [s] = 7800 Kg/m3

  • Configuration 3- Search for a symmetric sizing reinforcement:

  • calculation at EC2

  • FERR_COMP = “OUI “

  • FERR_SYME = “OUI “

  • SEUIL_SYME = 10 cm2

  • upper coating of csup = 4 cm

  • lower cinf coating = 4 cm

  • characteristic strength of concrete fck= 35 MPa

  • authorized limit stress in steel: μs, lim = 400 MPa

  • maximum compressive stress authorized at the level of the underside of concrete:

S-C, INF, LIM = 21 MPa

  • maximum compressive stress authorized at the level of the underside of concrete:

S-C, SUP, LIM = 21 MPa

  • steel-concrete equivalence coefficient: αE = 21

  • density of the steel [s] = 7800 Kg/m3

  • Setup 4- Calculation at BAEL91

  • calculation at BAEL91

  • FERR_COMP = “OUI “

  • upper coating of csup = 4 cm

  • lower cinf coating = 4 cm

  • characteristic strength of concrete FCj= 35 MPa

  • authorized limit stress in steel: μs, lim = 400 MPa

  • maximum compressive stress authorized at the level of the underside of concrete:

S-C, INF, LIM = 21 MPa

  • maximum compressive stress authorized at the level of the underside of concrete:

S-C, SUP, LIM = 21 MPa

  • steel-concrete equivalence coefficient: N = 21

  • density of the steel [s] = 7800 Kg/m3

1.4.3. Settings at the ELS Quasi-Permanent#

At ELS Permanent4 configurations will be considered on which we will start the calculation of the 10 load cases presented in §1.3.3:

  • Setup 1- Basic calculation:

  • calculation at EC2

  • upper coating of csup = 4 cm

  • lower cinf coating = 4 cm

  • characteristic strength of concrete fck= 35 MPa

  • elastic limit of steel: fyk = 500 MPa

  • Young’s modulus of steel: Eys = 210,000 MPa

  • maximum crack opening allowed on the underside: wmax, inf = 0.15 mm

  • maximum crack opening allowed on the upper side: wmax, sup = 0.15 mm

  • loading time coefficient: kt= 0.6

  • maximum compressive stress allowed in concrete for the control of non-linear creep: bsc, lim, NL= 15.75 MPa

  • estimated diameter of the bars on the lower face along the X axis: Φ INF, X = 25 mm

  • estimated diameter of the bars on the upper face along the X axis: Φ SUP, X= 25 mm

  • estimated diameter of the bars on the lower face along the Y axis: Φ INF, Y= 25 mm

  • estimated diameter of the bars on the upper face along the Y axis: Φ SUP, Y= 25 mm

  • steel-concrete equivalence coefficient: αE = 21

  • density of the steel [s] = 7800 Kg/m3

  • Configuration 2- Taking into account compression reinforcement:

  • calculation at EC2

  • FERR_COMP = “OUI “

  • upper coating of csup = 4 cm

  • lower cinf coating = 4 cm

  • characteristic strength of concrete fck= 35 MPa

  • elastic limit of steel: fyk = 500 MPa

  • Young’s modulus of steel: Eys = 210,000 MPa

  • maximum crack opening allowed on the underside: wmax, inf = 0.15 mm

  • maximum crack opening allowed on the upper side: wmax, sup = 0.15 mm

  • loading time coefficient: kt= 0.6

  • maximum compressive stress allowed in concrete for the control of non-linear creep: bsc, lim, NL= 15.75 MPa

  • estimated diameter of the bars on the lower face along the X axis: Φ INF, X = 25 mm

  • estimated diameter of the bars on the upper face along the X axis: Φ SUP, X= 25 mm

  • estimated diameter of the bars on the lower face along the Y axis: Φ INF, Y= 25 mm

  • estimated diameter of the bars on the upper face along the Y axis: Φ SUP, Y= 25 mm

  • steel-concrete equivalence coefficient: αE = 21

  • density of the steel [s] = 7800 Kg/m3

  • Configuration 3- Search for a symmetric sizing reinforcement:

  • calculation at EC2

  • FERR_COMP = “OUI “

  • FERR_SYME = “OUI “

  • SEUIL_SYME = 10 cm2

  • upper coating of csup = 4 cm

  • lower cinf coating = 4 cm

  • characteristic strength of concrete fck= 35 MPa

  • elastic limit of steel: fyk = 500 MPa

  • Young’s modulus of steel: Eys = 210,000 MPa

  • maximum crack opening allowed on the underside: wmax, inf = 0.15 mm

  • maximum crack opening allowed on the upper side: wmax, sup = 0.15 mm

  • loading time coefficient: kt= 0.6

  • maximum compressive stress allowed in concrete for the control of non-linear creep: bsc, lim, NL= 15.75 MPa

  • estimated diameter of the bars on the lower face along the X axis: Φ INF, X = 25 mm

  • estimated diameter of the bars on the upper face along the X axis: Φ SUP, X= 25 mm

  • estimated diameter of the bars on the lower face along the Y axis: Φ INF, Y= 25 mm

  • estimated diameter of the bars on the upper face along the Y axis: Φ SUP, Y= 25 mm

  • steel-concrete equivalence coefficient: αE = 21

  • density of the steel [s] = 7800 Kg/m3

  • Setup 4- Calculation at BAEL91

  • calculation at BAEL91

  • FERR_COMP = “OUI “

  • upper coating of csup = 4 cm

  • lower cinf coating = 4 cm

  • characteristic strength of concrete fck= 35 MPa

  • elastic limit of steel: fyk = 500 MPa

  • Young’s modulus of steel: Eys = 210,000 MPa

  • maximum crack opening allowed on the underside: wmax, inf = 0.15 mm

  • maximum crack opening allowed on the upper side: wmax, sup = 0.15 mm

  • loading time coefficient: kt= 0.6

  • maximum compressive stress allowed in concrete for the control of non-linear creep: bsc, lim, NL= 15.75 MPa

  • estimated diameter of the bars on the lower face along the X axis: Φ INF, X = 25 mm

  • estimated diameter of the bars on the upper face along the X axis: Φ SUP, X= 25 mm

  • estimated diameter of the bars on the lower face along the Y axis: Φ INF, Y= 25 mm

  • estimated diameter of the bars on the upper face along the Y axis: Φ SUP, Y= 25 mm

  • steel-concrete equivalence coefficient: αE = 21

  • density of the steel [s] = 7800 Kg/m3